Projectively flat Finsler manifolds with infinite dimensional holonomy

TL;DR

This paper investigates the holonomy groups of certain Finsler manifolds, demonstrating that projectively flat Randers and Bryant-Shen manifolds with non-zero constant flag curvature possess infinite dimensional holonomy groups, extending previous results.

Contribution

It proves that specific classes of projectively flat Finsler manifolds with non-zero constant flag curvature have infinite dimensional holonomy groups, advancing understanding of their geometric properties.

Findings

Holonomy groups of these manifolds are infinite dimensional.

Projectively flat Randers and Bryant-Shen manifolds have infinite holonomy.

Extends previous work on holonomy of Finsler manifolds.

Abstract

Recently, we developed a method for the study of holonomy properties of non-Riemannian Finsler manifolds and obtained that the holonomy group can not be a compact Lie group, if the Finsler manifold of dimension $> 2$ has non-zero constant flag curvature. The purpose of this paper is to move further, exploring the holonomy properties of projectively flat Finsler manifolds of non-zero constant flag curvature. We prove in particular that projectively flat Randers and Bryant-Shen manifolds of non-zero constant flag curvature have infinite dimensional holonomy group.